## Abstract

Subwavelength imaging can be obtained with alternately layered metallodielectric films structure, even when the permittivity of metal and dielectric are not matched. This occurs as the effective transversal permittivity tends to be zero or the vertical one approaches infinity, depending on the permittivity value of the utilized dielectric and metal material. Evanescent waves can be amplified through the structure, but not in a manner of fully compensating the exponentially decaying property in dielectric. Numerical illustration of subwavelength imaging is presented for variant configuration of anisotropic permittivity with finite layer number of metallodielectric films.

©2008 Optical Society of America

## 1. Introduction

It is first proposed and deduced mathematically by Pendry [1] that perfect lens can be obtained with a thin slab of negative refraction media (NRM) [2], which displays negative permittivity and negative permeability simultaneously. But the formidable challenge of fabricating homogenous NRM structures greatly hampers the application of perfect lens, especially in the optical frequency regime. Super lens, also proposed by Pendry, seems to be much appreciated due to the only requirement of negative permittivity and the fact that many noble metals, such as Ag, Au etc., display negative permittivity in a wide frequency range. Hence many investigations have been attributed to optical super lens theoretically [3–6] and experimentally [7, 8]. The condition for a good super lens with high resolution requires that the permittivity of metal and surrounding dielectric material is opposite to each other [1]. The mismatch of permittivity usually results in the considerable decrease of imaging resolution [3]. So it is hard to construct a super lens with appropriate materials at optical frequency of interest. Recently, the concept of super lens is further extended to the structure comprising stacked metal and dielectric films [8–11, 13–18]. Ramakrishna et al. points that improved subwavelength imaging with released resolution sensitivity to metal absorption can be obtained by cascading a great number of super lenses with very small thickness [8]. Hyper lens, which resembles Ramakrishna’s lens but in a cylindrical profile, is proposed [9, 10] and experimentally proved [11] to magnify objects beyond the diffraction limit. Most investigations are based on the no cut-off property of evanescent electromagnetic waves propagation in anisotropic media constructed with metallodielectric films, in which the effective permittivity in the transversal and normal directions have opposite signs [12, 13]. For instance, multiple subwavelength images in the specified direction can be observed in multi metallodielectric films due to the directed propagation of evanescent waves [13]. The metallodielectric films structure also provides a method to construct super lens when the metal and dielectric’s permittivity do not meet Pendry’s imaging condition. Pavel A. Belov et al. presents subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime [16]. Reference [17] argues that multilayer cascaded by alternative metal and insulator film with optimized geometrical parameters can also offer subwavelength imaging at wavelength their permittivity is greatly unmatched. The method proposed in reference [18] is different to the above investigations. The specific phase modulation effect in metal-dielectric-metal films structure with variant gap width is employed for imaging with subwavelength resolution.

In this paper, we mainly investigate subwavelength imaging with layered metal-dielectric films structure, especially when the permittivity of metal and dielectric are not matched. First, propagation of light in anisotropic structure and potential capability of subwavelength imaging are discussed in the approximation of effective material theory (EMT). Followed this is the analysis and comparison of amplification of evanescent waves for variant subwavelength imaging configuration of effective anisotropic permittivity. In the end, numerical illustrations of subwavelength imaging with finite layer numbers of metallodilectric films are presented and analyzed with optimal parameters.

## 2. Anisotropic media constructed by alternately layered dielectric and metal film

In this section, a brief review is presented for the optical characteristics of the effective anisotropic media, constructed by alternately and periodically stacked dielectric and metal film with tunable filling factor (see Fig. 1). For very thin stratified film, the effective medium theory can present good guidance and description of its optical behavior. The effective permittivity for layered film in the direction parallel and normal to the film are *ε _{x}*=

*ε*=

_{y}*ε*+

_{d}f*ε*(1-

_{m}*f*) and

*ε*

^{-1}

*=*

_{z}*ε*

^{-1}

*+*

_{d}f*ε*

^{-1}

*(1-*

_{m}*f*), respectively.

*f*is the filling factor for dielectric film.

As would be seen in the following sections and also in some references [7–10], appropriate effective anisotropic parameters are critical for subwavelength imaging. To give a comprehensive and convenient visualization of the available permittivity by tuning the filling factor, Fig. 2 plots the permittivity in the Cartesian coordinates of *ε _{x}* and

*ε*

^{-1}

*. The effective permittivity for layered metal and dielectric media are positioned at the line defined by*

_{z}*ε*

^{-1}

_{z}*ε*+

_{d}ε_{m}*ε*=

_{x}*ε*+

_{d}*ε*. The available permittivity is confined by two ends which construct two hyperbolas of

_{m}*ε*

^{-1}

_{z}*ε*=1 in quadrant I and III. The dispersion relation of plane waves in anisotropic media is

_{x}*k*

^{2}

*/*

_{x}*ε*+

_{z}*k*

^{2}

*/*

_{z}*ε*=

_{x}*K*

^{2}

_{0}, where ${k}_{0}=\frac{2\pi}{\lambda}$ ,

*k*and

_{x}*k*represent the wave vector. Shown in Fig. 3 are the representative relations between

_{z}*k*and

_{x}*k*in the different quadrants with variant permittivity signs. In quadrant I,

_{z}*ε*>0 and

_{x}*ε*>0, plane wave with ${k}_{x}>\sqrt{{\epsilon}_{z}}{k}_{0}$ would be evanescent in the media. If

_{z}*ε*and

_{x}*ε*are negative simultaneously (quadrant III), the media behaves like an anisotropic metal and no propagating mode is found here. For another two quadrants, II and IV, it is dramatically different. The media is characterized with no cut off property if

_{z}*ε*and

_{x}*ε*have different signs. That is to say, plane wave with very large

_{z}*k*can propagate in it. To understand this point explicitly, the structure can be viewed as strong anisotropic material which displays metal and dielectric property in the transversal and normal direction, respectively. Thus electromagnetic wave in the form of surface plasmon with very large

_{x}*k*can propagate inside this structure due to the coupling effect between any two adjacent layers of metal and dielectric films.

_{x}For the no cut off anisotropic media in quadrant II and IV, an interesting phenomenon should be particularly noted. The light nearly propagates in a fixed direction $\theta =\mathrm{atan}\left(\sqrt{\frac{-{\epsilon}_{x}}{{\epsilon}_{z}}}\right)$ with respect to the normal direction of the film. Considering its no cutoff property, it is not surprising that the directed propagating light have ability to restore the subwavelength features of light source. As pointed in reference [13], multiple point images can be observed for a point source depending on the absorption of light in the metal. This feature, obviously, would deliver overlapping effect of image for expanded objects in the transversal dimension.

What we are interested in is the permittivity at the coordinates of *ε _{x}* and

*ε*

^{-1}

*where*

_{z}*θ*=0. This means that plane wave with large transversal wave vector

*k*propagates in the direction normal to the film. This seems to provide a promise for subwavelength imaging for object positioned closely to the anisotropic media’s surface. As depicted in the permittivity diagram, these points are denoted as A(

_{x}*ε*=0 and

_{x}*ε*

^{-1}

*>0), B(*

_{z}*ε*<0 and

_{x}*ε*

^{-1}

*=0), C(*

_{z}*ε*>0 and

_{x}*ε*

^{-1}

*=0), D(*

_{z}*ε*=0 and

_{x}*ε*

^{-1}

*<0)and O(*

_{z}*ε*=0 and

_{x}*ε*

^{-1}

*=0). Cases A and B are in correspondence with -*

_{z}*ε*>

_{m}*ε*and cases C and D with -

_{d}*ε*<

_{m}*ε*. The dispersion relation turns into

_{d}*K*=0 and ${k}_{z}=\sqrt{{\epsilon}_{x}{k}_{0}}$ respectively. It is worth to note that for most part of optical frequency range the former two cases can be readily satisfied.

_{z}## 3. OTF for layered dielectric-metal structure

In the last section, superficial conclusion is made that point to point imaging with resolution beyond the diffraction limit is possible for the effective anisotropic media configured with permittivity at the five points A, B, C, D and O, as depicted in Fig. 1. But is it the fact? What is the difference in their imaging performances? Which one is best for super resolution imaging with the fixed value of ε_{m} and ε_{d}? These questions can be clearly answered with the optical transfer function for effective anisotropic media.

The following discussion is based on the transmission equation for plane wave incident on an anisotropic slab media with finite depth *d*, which is written as

where *ε* is the permittivity of dielectric material surrounding the anisotropic slab and
$k{\prime}_{z}=\sqrt{\epsilon {k}_{0}^{2}-{k}_{x}^{2}}$
.

For the two cases B and C with *ε _{z}*=∞ and

*ε*=

_{x}*ε*+

_{m}*ε*, the asymptotic form of Eq. (1) becomes

_{d}For the two cases A and D, where *ε _{x}*=0 and
${\epsilon}_{z}=\frac{{\epsilon}_{m}{\epsilon}_{d}}{{\epsilon}_{m}+{\epsilon}_{d}}$
, the transmission for

*k*≫

_{x}*K*

_{0}turns to be,

The two transmission Eqs. (2) and (3) display similar form of the reciprocal decrease of a linear function of *k _{x}*. They promise the enhancement of evanescent waves in a broad range of

*k*, compared with of evanescent waves’ exponentially decaying behavior in vacuum. The great difference between Eq. (2) and Eq. (3) is the triangular function dependence with

_{x}*d*or the Fabry-Perot effect in the finite depth of anisotropic media C.

Clearly, only the condition on *ε _{x}*=0 or

*ε*=∞ does not promise the desired optical transmission for perfect imaging which requires

_{z}*t*(

*k*) being constant valued for all

_{x}*k*component, indicating that the exponentially decaying property of evanescent waves is fully compensated through the structure. But fortunately, it does give rise to the amplification of evanescent wave and helps to implement subwavelength imaging. It is interesting to scompare alternately stacked metal and dielectric film to a single slab of lossless metal with

_{x}*ε*imbedded in dielectric

_{m}*ε*. In this case, the evanescent wave can be amplified but in a much narrowed small

_{d}*k*region. But taking a much thinner metal slab results in greatly extended amplification region of

_{x}*k*. This would help to understand evanescent wave enhancement in the anisotropic media comprising very thin metal and dielectric films.

_{x}There are two cases where perfect imaging occurs. One is the anisotropic media at the origin O (-*ε _{m}*=

*ε*). Both analytical analysis and numerical calculation [8] show that large

_{d}*k*plane waves can be completely transmitted through the media. But it is worth to note that the perfect transmission does not holds for low valued

_{x}*k*plane waves. Another one happens with the Fabry-Perot resonance condition $\sqrt{{\epsilon}_{x}}{k}_{0}d=m\pi $ for

_{x}*ε*=∞ and

_{z}*ε*>0 (case C with -

_{x}*ε*<

_{m}*ε*). According to Eq. (2), perfect imaging can be obtained theoretically with the transmitted amplitude for all

_{d}*k*to be 1. In reference [16], this type of perfect imaging is attributed to canalization imaging. But the perfect imaging in this case only occurs at discrete value of media depth, unlike anisotropic media with Pendry’s imaging condition -

_{x}*ε*=

_{m}*ε*. Both the two types of perfect imaging bear the virtue that imaging characteristics are not affected by the surrounding media.

_{d}In the end of this section, we would try to answer the question which configuration of anisotropic media is optimal for unmatched metal and dielectric (-*ε _{m}*≠

*ε*). To make the comparison easier, Eqs. (1) and (2) are further simplified according to the permittivity signs of cases A, B, C and D. Another assumption is made that

_{d}*ε*≈

*ε*for the fact that the refractive index difference for variant optical dielectric materials is usually small.

_{d}$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\approx \frac{1}{1-\frac{\mid {\epsilon}_{x}\mid}{2\epsilon {k}_{x}d}}\approx \frac{1}{1-\frac{\mid {\epsilon}_{m}+{\epsilon}_{d}\mid}{2{\epsilon}_{d}{k}_{x}d}}$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\approx \frac{1}{1+\frac{{\epsilon}_{x}{k}_{x}d}{2\epsilon}}\approx \frac{1}{1+\frac{\left({\epsilon}_{m}+{\epsilon}_{d}\right)}{2{\epsilon}_{d}{k}_{x}d}}$$

Let’s first considering the general permittivity condition with -*ε _{m}*>

*ε*, which results in cases A(

_{d}*ε*=0 and

_{x}*ε*

^{-1}

*>0) and B(*

_{z}*ε*<0 and

_{x}*ε*

^{-1}

*=0). Comparison between Eq. (4) and Eq. (5) yields |*

_{z}*t*(

_{A}*k*≫

_{x}*K*

_{0})|>|

*t*(

_{B}*k*≫

_{x}*K*

_{0})|. If -

*ε*<

_{m}*ε*, the result is reversed with |

_{d}*t*(

_{D}*k*≫

_{x}*K*

_{0})|<|

*t*(

_{C}*k*≫

_{x}*K*

_{0})|. It seems that high transmission occurs in cases A and C, localized at the axis of quadrant I (anisotropic dielectric quadrant) in Fig. 2. In addition to the much lower capability of conversion of evanescent waves, cases B and D also surfer from the phase modulation which are not plotted. This usually results in reduced imaging property for the incorrect restoration of phase information.

To give further numerical illustration, Fig. 4 presents the OTF curves for typical dielectric permittivity ε_{d}=2 and three representative metal permittivity with ε_{m}=-5 (for cases A and B), ε_{m}=-1 (cases C and D), and ε_{m}=-2 (O). Shown in Fig. 4(a), the transmitted amplitude of evanescent waves decays exponentially with increasing *k _{x}* for anisotropic parameters localized in quadrant I and III (noting the OTF is in logarithmic scales). But for those in quadrant II, obvious oscillating OTF curves with FP effect can be observed. As for the four cases at permittivity diagram coordinates, OTF decrease in a reciprocal way of linear function, not obvious in logarithmic scale. If the permittivity of metal and dielectric are matched in case O, complete conversion of evanescent waves with large

*k*can be observed. Also can be clearly seen is case A displays superior performance than that of case B and case C much better than D, illustrating the result in the above analysis. According to Eq. (4), the resolution of the anisotropic imaging structure can be approximated as

_{x}*π*(

*ε*+

_{m}*ε*)

_{d}*d*/

*ε*by taking the position with half of the transmission function. So the best resolution occurs with matched metal and dielectric cases.

_{m}## 4. Subwavelength imaging with finite layered absorption metal and dielectric

In the last section, useful and instructive analysis is presented for the OTF of effective anisotropic permittivity constructed with layered lossless metal and dielectric media. Now we consider the practical cases for finite layer of stacking films with absorption metal. The purpose of this section mainly focuses on the illustration of the subwavelength imaging of the two cases of layered metal-dielectric media, cases A and D for unmatched dielectric and metal condition. The cases B and C are not considered here any more for their terrible OTF curves as indicated in the last section and numerically simulated images which are not presented.

Figure 5 gives the calculated OTF with transfer matrix method for the recommended anisotropic cases (A, C and O) with finite layers of dielectric and absorption metal. The imaginary permittivity of metal is assumed to be one tenth of its real part as an approximation for measured metal permittivity. The important phenomena is the OTF curves converge with increased number of layers to the one calculated by effective anisotropic parameters with Eq. (1). This fact promises the analysis in the above sections to be a good guidance. In comparison with Fig. 4, the OTF curves decrease much faster and nearly exponentially for large *k _{x}* now. This is mainly attributed to the absorption of light in metal. As shown in paper [8], stacking films of permittivity matched metal and dielectric greatly relieve the great deleterious effect of metal loss on the image resolution. Here, it is illustrated that the same effect happens for stacked metal and dielectric film with unmatched permittivity as well.

Assume an object of width 2*a* positioned closely at the interface of layered anisotropic structure and surrounding media as depicted in Fig. 1. Note the object is extended uniformly to infinity in the y direction and it can be decomposed into a series of plane waves as

where $\tilde{E}\left({k}_{x}\right)=\frac{\mathrm{sin}\left({k}_{x}a\right)}{{k}_{x}a}$ for rectangular spike objects.

At the other interface facing the object,

Figure 6 plots the calculated images for finite-numbered layer of metal and dielectric stacked films with parameters in correspondence with Fig. 5. The spike width 2*a*=0.1*λ*. Clearly, cases A, C and O all have the ability of imaging subwavelength objects, but display different imaging performance. The minimum full width at half maximum (FWHM) of *E _{x}* component image for case A is about 0.25

*λ*.

*E*Images in cases C and O are narrower, about 0.1

_{x}*λ*wide. The more layer numbers, the narrower image width with a convergence to the effective anisotropic material approximation. This point can also be seen from the OTF curves in Fig. 5. On the other hand, they all exhibit dramatically difference between imaging field

*E*and

_{x}*E*due to the vector nature of transversal magnetic polarized field. Usually, the width of

_{z}*E*field is greatly expanded compared with

_{z}*E*because

_{x}*E*is mainly localized at the edges of object. This would inevitably lower imaging resolution of electric field intensity and bring artifacts to images.

_{z}The deleterious influences also come from the resonant excitation of surface plasmon modes at some transversal wavevectors. This is dramatically paradoxical since surface plasmon is believed to be the physical origin of super resolution imaging [1]. In case A (Fig. 6(a) and Fig. 6(b)), this unwelcome effect is particularly obvious with greatly and transversally extended electric field as background. This can be well understood from the OTF curves where a transmission peak appeared at *k _{x}* slight larger than that in dielectric media, indicating the surface plasmon mode resonance. Similar effect can also be observed for case O with matched permittivity (Fig. 6(e) and Fig. 6(f)), but not so terrible like case A. The difference, we believe, arises from the great localization and propagation loss of surface plasmon in permittivity matched environment. This can also be justified through the OTF plots where the resonance peak is much damped and widened (Fig. 5(c)). Image in case C does not display this effect clearly. Here no transversal wave vector supports evanescent waves and they are all in propagating state (see Fig. 3). The resonant peak in Fig. 4(c) is the Fabry-Perot mode which does not bring obvious extension of images because light propagates in the direction almost normal to the films. Other reasons accounting to the reduced imaging property also include the phase modulation of OTF due to absorption and waveguide modes presence, which is not discussed here any more.

## 5. Conclusion

Anisotropic media constructed by alternately layered metal and dielectric films can support propagation of evanescent waves. This suggests that point to point subwavelength imaging is possible by taking the propagation direction normal to the stacked films, which requires *ε _{x}*=0 or

*ε*

^{-1}

*=0. OTF analysis with effective material theory shows transmitted amplitude of plane wave decrease in a reciprocal way of linear function with respect to increased transversal wave vector*

_{z}*k*. In addition, it is shown that condition

_{x}*ε*=0 appreciates metal with larger absolute permittivity than the dielectric and

_{x}*ε*

^{-1}

*=0 just the contrary. Not so good for perfect imaging with almost complete conversion of evanescent waves like super lens, but they do render the capability of subwavelength imaging as illustrated with numerical simulation for finite layer of absorption metal and dielectric films.*

_{z}## Acknowledgments

This work was supported by 973 Program of China (No.2006CB302900) and National Natural Science Foundation of China (No.60778018) and 863 Program of China (2006AA04Z310).

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